Separation Axioms T0 to T4: Topology Explained Simply

The separation axioms T0 through T4 form a ladder measuring how well a space separates points and closed sets with open neighborhoods. T0 only asks that distinct points be topologically distinguishable; T4 requires disjoint closed sets to fit inside disjoint open sets. Each level implies all levels below it.

They matter because analysis quietly assumes enough separation: sequence limits are unique only in Hausdorff (T2) spaces, and Urysohn's lemma, which builds continuous functions separating closed sets, needs normality (T4). Knowing which axiom holds tells you which tools are legal.

Key Concepts

• T0 (Kolmogorov): for any two distinct points, at least one has a neighborhood not containing the other.
• T1 (Frechet) each point has a neighborhood missing the other; equivalently, singletons are closed.
• T2 (Hausdorff): any two distinct points have disjoint neighborhoods.
• T3 (regular) T1 plus: every closed set and point outside it have disjoint neighborhoods.
• T4 (normal): T1 plus: any two disjoint closed sets have disjoint neighborhoods.
• Implication chain: T4 implies T3 implies T2 implies T1 implies T0, and none of the reverse implications holds in general.
• Metric spaces: every metric space is T4, hence satisfies all five axioms.

How It Works

To check an axiom, attempt the separation it demands and hunt for a counterexample when it fails. In metric spaces the half-distance trick does the work: for distinct x, y at distance d, the balls of radius d/2 are disjoint, proving the space is Hausdorff, and a similar argument with distances to closed sets proves normality. Subspaces of T0, T1, T2, and T3 spaces keep the property, but normality can be lost in subspaces. Products preserve T0 through T3, though products of normal spaces need not be normal. The cofinite topology (T1 but not T2) and the Sierpinski space (T0 but not T1) are the standard separating examples.

Worked Example

Prove R with the usual topology is Hausdorff. Take x = 1 and y = 4, distance d = 3. With radius r = d/2 = 1.5, the neighborhoods U = (1 - 1.5, 1 + 1.5) = (-0.5, 2.5) and V = (4 - 1.5, 4 + 1.5) = (2.5, 5.5) are disjoint, since everything in U is below 2.5 and everything in V is above 2.5. The same half-distance construction works for any two distinct reals, so R is T2. The argument uses only the metric, so every metric space is Hausdorff; with distances to closed sets one shows metric spaces are even T4.

Common Misconceptions

• "T3 means regular, period." Conventions vary: many authors define T3 as regular plus T1, while some use "regular" alone. Always check whether the author folds T1 into the definition.
• "Every reasonable space is Hausdorff." The Zariski topology in algebraic geometry and the cofinite topology are T1 but not T2, and they are perfectly reasonable in their contexts.
• "Subspaces of normal spaces are normal." Normality is not hereditary: a subspace of a T4 space need not be T4, though closed subspaces do inherit normality.

Real-World Applications

• Analysis uniqueness of sequence limits and of continuous extensions needs the Hausdorff property.
• Urysohn's lemma: in a normal (T4) space, disjoint closed sets are separated by a continuous real-valued function, the basis for partitions of unity.
• Manifold theory: topological manifolds are required to be Hausdorff (and second countable) so that calculus on them behaves sensibly.

The separation axioms form a strict hierarchy from T0 to T4, with metric spaces at the top. Learn the implication chain, the half-distance proof, and counterexamples like the cofinite topology, and every separation hypothesis in a theorem becomes readable.